1 / 29

Alpha Recycling in Confirmatory Clinical Trials

Unifies, Simplifies, Extends many common MTPs. Alpha Recycling in Confirmatory Clinical Trials. Olivier Guilbaud Senior Principal Scientist AstraZeneca R&D, Södertälje, Sweden. Outline . Background. Splitting, Recycling, Adding, (parts of alpha). Graphs and Default Graphs for MTPs .

kin
Download Presentation

Alpha Recycling in Confirmatory Clinical Trials

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Unifies, Simplifies, Extendsmany common MTPs Alpha RecyclinginConfirmatory Clinical Trials Olivier GuilbaudSenior Principal ScientistAstraZeneca R&D, Södertälje, Sweden

  2. Outline • Background • Splitting, Recycling, Adding, (parts of alpha) • Graphs and Default Graphs for MTPs • Rejection Algorithm • Holm’s MTP for groups of Hs, and a problem to discuss • Improvements through extensions of the Default Graph • Summary, and further results: • (Simultaneous Confidence regions, CTP-formulation of MTPs)

  3. Background: Regulatory environment Citation from EMEA/CPMP’s (2002) Points to Consider onMultiplicity Issues in Clinical Trials : “ … multiplicity can have a substantial influence on the rate of false positive conclusions …” ”As a general rule it can be stated that control of the family-wise type-I error rate in the strong sense… is a minimal prerequisite for confirmatory claims”

  4. Several Efficacy Variables • Several Tolerability Variables • Several Comparators (e.g. Placebo, Active 1, Active 2, …) • Several Doses • Several kinds of Administration (e.g. once daily, twice daily, …) • Delta-Noninferiority / Superiority / Delta-Superiority • Several Subgroups/Kinds of Subjects Background: Multiple Confirmatory Comparisons Confirmatory: Pre-specified family of multiple comparisons (i.e. statistical tests) . Risk of getting any ”false positive” result/conclusion is  (no matter how many or which H0s are true – Strong Control)

  5.   ,no matter how many, or which, His in the family are true Background: Abbreviations/Terminology • MTP : Multiple Testing Procedure • MCP : Multiple Comparison Procedure • FWER : Family-Wise Error Rate (type-I errors) = • Pr[at least 1 trueHi in the Family is rejected by the MTP] • (to be controlled to be   in the strong sense) In contrast to the weak sense (old) :  , if allHis in the family are true

  6. Fixed-Sequence MTP (old) p1  , p2  , p3   • Fallback MTP (2003/2005)weights w1, w2, w3p1  w1, p2  (w2+), p3  (w3+) Background: brief refresher about three basic MTPs Three basic MTPs for Family {H1, H2, H3} based on raw p-values p1, p2, p3 • Holm MTP (1979)p(1)  /3, p(2)  /2, p(3)  /1 Now, to the Alpha-Recycling framework

  7. Only this now Two articles in Statistics in Medicine, February 2009

  8. Bonferroni: split of  Fixed-Sequence: recycling of  Combination of split & recycling of  Idea: Splitting and Recycling test-mass (= part of ) Throughout: Raw p-value available for each null hypothesis H

  9. Parallel-gatekeeping MTP: 2 graphs for same MTP Default graph Important for rejection Algorithm Its sequences reflect the possible rejection paths Idea: Adding test-mass from different paths

  10. Fallback MTP for 3 Hs: 2 graphs for same MTP Default graph Important for rejection Algorithm Its sequences reflect the possible rejection paths Idea: Adding test-mass from different paths

  11. Holm’s MTP for 3 Hs: 2 graphs for same MTP Default graph Important for rejection Algorithm Its sequences reflect the possible rejection paths Idea: Adding test-mass from different paths

  12. Step 1. For each ”first”H in Sequences, • (a) add-upp test mass from ”its” Sequences(b) test H at this added-up level • If nothing is rejected, then Stop; otherwise: • Reduce Sequences by deleting all rejected H from Sequences, and go to Step 2 with remaining reduced Sequences • Step r = 2, 3, …. For each ”first”H in remaining reduced Sequences, • (a) add-upp test mass from ”its” Sequences(b) test H at this added-up level • If nothing is rejected, then Stop; otherwise: • Reduce Sequences by deleting all rejected H from Sequences, and go to Step r + 1 with remaining reduced Sequences Rejection Algorithm - based on Default-Graph Seqs This MTP controls (in the strong sense) the FWER to be   !

  13. Comments • number of sequences can be increased(no restriction) • length of sequences can be increased(no restriction) • if not all sequences contain all Hs, then improvements are possible (by increasing length of relevant sequences) • interpretability of MTP is important(also when improvements are considered)

  14. /2 /2 Bonferroni for 2 groups of 2 Hs, with Fixed-Seq testing within group 1 2 3 4 Holm for 2 groups of 2 Hs, with Fixed-Seq testing within group: /2 /2 1 1 2 3 4 /2  1 1 2 3 4 A refresher – Bonferroni & Holm for groups of Hs Step1: Bonferroni-version as above Step2: If one group is entirely rejected, but not the other, then try again in the other at increased level 

  15. Bonferroni for 2 groups of 2 Hs, with Fixed-Seq testing within group /2 /2 1 2 (is already in default-graph form): 3 4 Extend sequences of default graph to get recycling improvement equivalent toHolm for 2 groups of 2 Hs, with Fixed-Seq testing within group /2 /2 1 2 3 4 Problem to discuss with your neighbour

  16. H1 w (1-w) H2 H3 H4 Recent Example discussed by FDA statisticians • Confirmatory family with 4 null hypotheses :

  17. Alpha-recycling applied to show Non-inf & Superiority for Primary & Secondary Variables Basic version Reformulation in Recycling terms w  (1-w) H1 H1 H1 w (1-w) H2 H3 H2 H3 H4 H4 Recent Example discussed by FDA statisticians Very easy to test with rejection algorithm ! But why not recycle also after the last H in each sequence ?

  18. Improvement Reformulation in Recycling terms w (1-w)  H1 H1 H1 w (1-w) H2 H3 H2 H3 H4 H3 H4 May be questionedby some (not me)! May be questionedby some (not me)! H4 H2 Recent Example discussed by FDA statisticians Alpha-recycling applied to show Non-inf & Superiority for Primary & Secondary Variables - Simple improvement Very easy to test with rejection algorithm ! This improved MTP is ”-exhaustive” !

  19. Improved Fallback MTP for 3 Hs: 2 graphs for same MTP Default graph Sequences can be extended/added for more rejections This improved MTP is ”-exhaustive” !

  20. Improved Parallel-gatekeeping : 2 graphs for same MTP Default graph Sequences can be extended/added for more rejections This improved MTP is ”-exhaustive” !

  21. Reformulation in Recycling terms Original proposal Default graph  to show strong control of FWER 1 4 2 3 5 6 7 8 9 11 10 12 /8 /8 /8 /8 /8 /8 /8 /8 1 1 4 4 1 1 4 4 2 3 5 6 2 3 5 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 11 11 11 11 10 10 10 10 12 12 12 12 XXX example: 2 primary & 2 second vars, 2 age stata Simple improvement possible (  Holm instead of Bonf after H8 ): Add sub-seq (11, 12) to Default-graph sequences ending with (9, 10) Add sub-seq (9, 10) to Default-graph sequences ending with (11, 12)

  22. Reformulation in Recycling terms Original proposal Default graph  1 4 2 3 5 6 7 8 9 11 10 12 /8 /8 /8 /8 /8 /8 /8 /8 1 1 4 4 1 1 4 4 2 3 5 6 2 3 5 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 11 11 11 11 10 10 10 10 12 12 12 12 Compact notation for MTP in XXX example Compact notation for this recycling MTP(B here is Bonf-operator): B(H1B(H2, H3), H4B(H5, H6)) H7H8B(H9H10, H11H12) Can be expanded (using certain rules) to 8 sequences of Hs above

  23. Reformulation in Recycling terms Basic version Default graph /4 /4 /4 /4 /2 /2 1 1 2 2 1 2 3 5 5 4 3 5 4 6 6 6 YYY example: 3 doses vs. 0, 1 primary & 1 second var Idea: To give lowest dose a chance if at least 1 of the 2 larger doses works for the primary variable

  24. Reformulation in Recycling terms Improvement 1 Default graph /4 /4 /4 /4 /2 /2 1 1 2 2 1 2 3 5 5 4 3 5 4 6 6 5 5 6 6 6 Increases level for H5 and H6 if H3 and/or H4 are rejected YYY example – Improvement 1

  25. Reformulation in Recycling terms Improvement 2 Default graph  /4 /8 /8 /8 /8 /4 1 1 1 2 2 2 1/2 1/2 1 2 3 5 5 5 5 4 3 5 4 6 6 6 6 5 5 6 6 2 2 1 1 6 Gives H1 or H2 a (Holm-type) second opportunity to be rejected if not rejected initially 2 3 4 4 3 1 4 4 3 3 4 3 YYY example – Improvement 2

  26. Summary, and further results • Easy to formulate (cf. Closed-Testing Pocedures) • Easy to perform (even manually) • Unifies many common MTPs • Easy to construct new MTPs (for new problems) • Easy to calculate multiplicity-adjusted p-values (algorithm similar to that for rejections) • Compact algebraic notation available(to describe and derive default graphs) • Easy to obtain weights of Bonferroni test for any intersection hypothesis HI of corresponding CTP (useful for contruction of Conf regions)

  27. Fixed-Sequence MTP (old) p1  , p2  , p3   • Fallback MTP (2003/2005)weights w1, w2, w3p1  w1, p2  (w2+), p3  (w3+) Corresponding Simultaneous Confidence Regions ??? Three basic MTPs for Family {H1, H2, H3} based on raw p-values p1, p2, p3 Solution inJASA 1999 • Holm MTP (1979)p(1)  /3, p(2)  /2, p(3)  /1 Open until2007

  28. Amazing: Simultanouslypresented at MCP 2007in Vienna 28 years after Holm (1979) ExtensionsandRelations Corresponding Simultaneous Confidence Regions !!! Confidence regions require weights that can be obtained directly from rejection algorithm of recycling approach

  29. Outline • Background • Splitting, Recycling, Adding, (parts of alpha) • Graphs and Default Graphs for MTPs • Rejection Algorithm • Holm’s MTP for groups of Hs, and a problem to discuss • Improvements through extensions of the Default Graph • Summary, and further results: • (Simultaneous Confidence regions, CTP-formulation of MTPs)

More Related